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Banach Center Publications

2000 | 53 | 1 | 199-204
Tytuł artykułu

Reduction of differential equations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) $Ly = D^{n}y + a_{n-1}D^{n-1}y+...+ a_{0}y = 0$, where $a_0,... ,a_n ∈ F$, and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system $y_1,... ,y_n$ of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms of E leaving the elements of F fixed. If E = PV(F,Ly=0) is a Picard-Vessiot extension, then the elements g ∈ G(E|F) are n × n matrices, n= ord L, with entries from C, the field of constants. Is it possible to solve an equation (1) by means of linear differential equations of lower order ≤ n-1? We answer this question by giving neccessary and sufficient conditions concerning the Galois group G(E|F) and its Lie algebra A(E|F).
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
199-204
Opis fizyczny
Daty
wydano
2000
Twórcy
autor
• Institute of Mathematics, Polish Academy of Sciences, Katowice Branch, Bankowa 14/343, 40-007 Katowice, Poland
autor
• Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany
Bibliografia
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• E. L. Ince [1] Ordinary differential equations, Dover Publ., New York, 1956.
• Y. Kaplansky [1] An introduction to differential algebra, Hermann, Paris, 1976.
• E. R. Kolchin [1] Algebraic matrix groups and Picard-Vessiot theory of homogeneous linear ordinary differential equations, Ann. of Math. 49 (1948), 1-42.
• E. R. Kolchin [2] Differential algebra and algebraic groups, Academic Press, New York, 1973.
• S. Lang [1] Algebra, Reading, Addison-Wesley Publ., 1984.
• A. R. Magid [1] Lectures on differential Galois theory, American Math. Soc., 1994.
• J. Mikusiński [1] Operational calculus, Pergamon Press, New York, 1959.
• M. F. Singer [1] Solving homogeneous linear differential equations in terms of second order linear differential equations, Amer. J. Math. 107 (1985), 663-696.
• M. F. Singer [2] Algebraic relations among solutions of linear differential equations: Fano's theorem, Amer. J. Math. 110 (1988), 115-144.
• M. F. Singer [3] An outline of differential Galois theory, in: Computer algebra and differential equations, E. Tournier (ed.), Academic Press, 1989, 3-57.
• K. Skórnik and J. Wloka [1] Factoring and splitting of s-differential equations in the field of Mikusiński, Integral Transforms and Special Functions 4 (1996), 263-274.
• K. Skórnik and J. Wloka [2] m-Reduction of ordinary differential equations, Coll. Math. 78 (1998), 195-212.
• K. Skórnik and J. Wloka [3] Some remarks concerning the m-reduction of differential equations, Integral Transforms and Special Functions 9 (2000), 75-80.
• C. Tretkoff and M. Tretkoff [1] Solution of the inverse problem of differential Galois theory in the classical case, Amer. J. Math. 101 (1979), 1327-1332.
• J. T. Wloka [1] Über lineare s-Differentialgleichungen in der Operatorenrechnung, Math. Ann. 166 (1966), 233-256.
• A. E. Zalesskij [1] Linear groups, Encycl. of Math. Sciences 37 (Algebra IV), Springer, Berlin, 1993.
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Bibliografia
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